Every sequence has a convergent subsequence?

I'm not sure if this is true or not. but from what I can gather, If the set of Natural numbers (divergent sequence) {1, 2, 3, 4, 5,...} is broken up to say {1}, is this a subsequence that converges and therefore this statement is true?

You are wondering if it is always possible to find at least one convergent subsequence in any sequence however divergent?
Your specific question amounts to whether a sequence with a finite number of elements is convergent.
To answer that, check the definition of "converge".

Does {1,2,3,..,n,.. } , i.e., a_n:=n have a convergent subsequence? Of course, this depends on your topology, but, as
a subspace of the Reals, does this have a convergent subsequence? This is one of the characterization of compact metric spaces, as every sequence having a convergent subsequence. And, in the subspace topology of the Reals,
{1,2,3,....} is/not compact (find a cover by open sets so that each contains a single number).